I think it depends on the age of your child.
When the kids were much younger we took our own because we wanted it ALL the time (throught the airport, shopping, walking back to the tram, monorail, car). But I hated having to fold it up and lug it on and off of things and worrying if it went missing because a CM had moved it, but I thought it had been taken.
Now we rent a double. They are expensive, but we just figure it into the cost of our trip, considering how much you spend to go to Disney it isn't that much of the budget. The kids can walk to and from things and around the mall, but the parks are too much for them. Those strollers are the BEST. They ride so smoothly, we get a double and both kids can get in. The back has a place for a cup or two and we just hang our bag on the handle or put it on top of the shade. We try to travel light. I would rather budget extras into my trip and pay for the convenience of not lugging things around the hot park, especially when my feet hurt. Now I am just trying to figure something great for distinguishing our rented strollers from others that can easily be removed at the end of the day when we return it.